Abstract

On the basis of the linearized theory of water waves, the present study has demonstrated a semi-analytical method to assess the hydrodynamic performance of a pair of partially immersed barriers just above a thick bottom-standing barrier. By means of the eigenfunction expansion method, a system of the first kind Fredholm-type integral equation involving a horizontal component of velocity as unknown functions is developed for the interaction of water waves with both types of barriers. The multiterm Galerkin approximation is adopted to determine these unknown functions having square root singularities at the submerged edge of the thin barriers and one-third singularities at the corners of the thick barrier. In order to overcome such types of singularities, Chebychev polynomials for half-singularities and ultra-spherical Gegenbauer polynomials for one-third singularities with suitable weight functions have been taken into consideration. The numerical examples of both reflection and transmission coefficients are presented to examine the hydrodynamic performance of breakwater. Some fascinating results like resonant frequencies are obtained for practical engineering. At the same time, reflection coefficients for the present breakwater agree reasonably for the limiting cases with the previously available results.

Introduction

In the past few decades, the vertical breakwater has paid considerable attention for its profound impact on the mitigation of the wave reflection. Vertical breakwaters are extensively used in ocean engineering to defend ports and operation areas for ship loading and unloading. These breakwaters also have provided a protection for the coastal regions from the intrusion of littoral drift. By utilizing the vertical barriers, an extremely stable structure has developed even in rough, deep seas and such cause leads to their use throughout the world.

Water wave scattering by vertical barriers becomes significant in the study of water wave problems from the last few decades. The scattering of water wave problems by appropriate arrangement of thin vertical barriers in finite depth water or deep water has been carried out by a large number of researchers, in the literature within the framework of the linear theory of water waves. In the context of vertical barriers, the early works of Ursell [1,2] motivated many researchers to deal with the problems of scattering and radiation of water waves by vertical barriers. Dean [3] exercised the complex variable method to encounter the scattering problem by the submerged plane barrier. Meanwhile, many scientists and researchers have devoted their efforts to improve different methods from time to time to reveal these problems in deep water, for example, Stoker [4], Levine and Rodemich [5], Lewin [6], and Mei [7]. Also, some numerical and semi-analytical methods have been applied to solve the problems in finite depth water such as Liu and Abbaspour [8], Lasoda et al. [9], Abul-Azm [10], and Porter and Evans [11].

In the offshore regions, the breakwaters in the form of thick barriers have increased over the past few decades. Water wave scattering by different structures of thick barriers has been investigated widely by the marine scientists and researchers. On the grounds of the huge sizes of these structures as compared to the wavelength of the ocean waves, the thickness of the configuration plays a major role in the analysis. The study of Mei and Black [12] is considered to be the pioneer work in water wave scattering by thick vertical barriers with a rectangular cross-section in uniform finite depth. With the assistance of the conformal mapping method, Guiney et al. [13] investigated the wave propagation problem by including the effect of thickness in a vertical wall of a rectangular cross-section. The problem of scattering of a normally incident surface wave train by an obstacle in the form of a thick vertical barrier of four different basic configurations was analyzed by Kanoria et al. [14] using the multiterm Galerkin technique involving ultra-spherical Gegenbauer polynomials. Later, Mandal and Kanoria [15] considered the oblique incident wave train to extend the work of Kanoria et al. [14]. Sturova [16] introduced an efficient tool for scrutinizing the behavior of long surface waves generated by periodic disturbances in a fluid with an uneven bottom of a particular type.

In the recent decades, analysis of both half and one-third singularities in the integral equations that arise in the problems of water waves is a matter of contention among the researchers. Ray et al. [17] studied wave propagation over submarine trench in the presence of a thin vertical partially immersed barrier for its three different positions. They discussed different types of integrable singularities (square root and cube root) at the end points for solving the integral equations. Ray et al. [18] also used simple polynomials as basis functions to tackle singularities in the problem of wave propagation over an infinite step in the presence of a thin vertical barrier of four different geometrical configurations. Recently, Sarkar et al. [19] endowed the literature of water waves by choosing Chebychev and Gegenbauer polynomials as basis functions to handle both half and one-third singularities.

The study of wave diffraction by the combination of thick and thin barriers using the semi-analytical method (solution of integral equations involving two types of singularities) is quite new in this field of research. This study is, first of all, about to give a promising procedure for encountering wave interaction with a pair of thin vertical partially immersed barriers between the corners of a bottom-standing thick barrier, and second, about to provide a reliable benchmark for complicated numerical solutions of integral equations having integrable singularities (half and one-third singularities). We have investigated the problem of wave interaction with a pair of partially immersed barriers in the presence of a rectangular thick bottom-standing barrier. So far as our knowledge goes these configurations of the problems have not been investigated in the literature. With the aid of Havelock’s [20] inversion formulae, the problem is reduced to a four first kind Fredholm-type integral equations involving the horizontal components of velocity as unknown functions above the corners of thick barrier and below the submerged edge of the thin barriers. The methodology of multiterm Galerkin approximation along with Chebychev and Gegenbauer polynomials has been employed to tackle half and one-third singularities near the submerged edges of the thin barriers and corners of thick barrier, respectively. To verify the present numerical model, the results of Das et al. [21] for thin vertical barriers without thick bottom-standing barrier and Mandal and Kanoria [15] for the thick bottom-standing barrier without thin vertical barriers are validated. These analyses reveal that the present simulation and the researches of Das et al. [21], Mandal and Kanoria [15] fall into a good agreement. Based on numerical results for reflection and transmission coefficients, this model shows to be useful and can be adequately brought into play as an applicable tool in marine engineering.

Theoretical Development

The detail of the theoretical background has been presented in Fig. 1, where two unequal thin vertical partially immersed barriers of lengths c2, c3 are situated at x = s2, s3, respectively, between the corners at x = s1, s4 of the thick bottom-standing. The y-axis is directed vertically downward, and the xz-plane is considered to be the equilibrium water surface. The whole fluid region is assumed to be incompressible, nonviscous, and the flow irrotational. So that the fluid may be represented in terms of velocity potential Re[ϕinc(x,y)eiτziωt], where Re[] denotes the real part of a complex expression and
ϕinc(x,y)=coshk0(hy)coshk0heiκ(xs1)
with τ = k0 sinα, κ = k0 cosα, α is the angle of inclination and k0 is the unique real positive root in the regions Ω1 and Ω5 of the dispersion equation ktanh kh = K, K = ω2/g, ω is the angular frequency of the incoming wave train, g be the gravitational acceleration.
Fig. 1
Schematic diagram of the problem
Fig. 1
Schematic diagram of the problem
Close modal
According to Snell’s law for refraction across discontinuities in the fluid depth, τ satisfies (cf. Kirby and Dalrymple [22]).
τ=k0sinα=α0sinα=constant
where α0 is the wave number for the region Ω2, Ω3, and Ω4 and also the unique real positive root of the equation α″tanhαc = K (here c = c1 = c4, gap between the free surface and the top of the thick barrier).
The velocity potential ϕj(x, y) satisfies the modified Helmholtz’s equation in each subregion Ωj,
2ϕj=τ2ϕjinΩj(j=1,2,5)
(1)
The condition at the free surface is given by
Kϕj+ϕjy=0ony=0,<x<
(2)
The conditions on the two sides of the thick barrier and on the thin barrier are
ϕj+1x=0on{x=sj,cj<y<h(j=1,4)x=sj,0<y<cj(j=2,3)
(3)
The bottom boundary condition is given by
ϕjy=0on{y=h,x<s1,x>s4(j=1,5)y=c,s1<x<s4(j=2,3,4)
(4)
The conditions at the corners of the thick barrier and the submerged edge of the thin barriers are given by (cf. Mandal and Chakrabarti [23])
r1/3ϕjisboundednearthecornersofthethickbarrier
(5a)
and
r1/2ϕjisboundednearthesubmergededgeofthinbarriers
(5b)
Also, the infinity condition is given by
ϕ(x,y){ϕinc(x,y)+Rϕinc(x,y)asxTϕinc(x,y)asx
(6)
where R and T are reflection and transmission coefficients.
Using the eigenfunction expansion method, the general solution for potential function ϕj(x, y) in each of the five subdomains is given by
ϕ1(x,y)={eiκ(xs1)+V1eiκ(xs1)}coshk0(hy)coshk0h+n=1Rn(1)coskn(hy)cosknhepn(xs1)inΩ1
(7a)
ϕj(x,y)={R0(j)cosq0(xsj1)+T0(j)sinq0(xsj)}coshα0(cy)coshα0c+n=1{Rn(j)coshqn(xsj1)+Tn(j)sinhqn(xsj)}cosαn(cy)cosαncinΩj(j=2,3,4)
(7b)
ϕ5(x,y)=V2eiκ(xs4)coshk0(hy)coshk0h+n=1Tn(5)coskn(hy)cosknhepn(xs4)inΩ5
(7c)
where ±ikn(n=1,2,) are purely imaginary roots of ktanh kh = K; ±iαn(n=1,2,) are of αtanhαc=K,pn=kn2+τ2, p0=k02τ2,qn=αn2+τ2, and q0=α02τ2. Here, Rn(j)(j=1,2,3,4),Tn(j)(j=2,3,4,5) are unknown constants; V1 and V2 are given by V1=Re2iκs1,V2=Teiκ(s4s1).
For the shake of forming integral equations, we define the horizontal components of velocity uj(y)(j = 1, 2, 3, 4) and difference of potentials vj(y) (j = 1, 2, 3, 4) as
uj(y)=ϕjx(sj0,y)=ϕj+1x(sj+0,y),0<y<h
(8)
and
vj(y)=ϕj+1(sj+0,y)ϕj(sj0,y),ylj
(9)
where
uj(y)=0for{cj<y<h,j=1,40<y<cj,j=2,3vj(y)=0for{0<y<cj,j=1,4cj<y<c,j=2,3lj={(0,h),j=1,4(0,c),j=2,3
The nature of horizontal components of velocity near the corners of the thick barriers and near the submerged edge of thin barriers are, respectively, given as
uj(y)=O(|ycj|1/3)asycj0(j=1,4)
and
uj(y)=O(|ycj|1/2)asycj+0(j=2,3)
Let us introduce the following constants:
X1=1,X2=0,X3=0,X4=0,Y1=V1,Y2=T0(3)sinq0(s3s2)Y3=T0(4)sinq0(s4s3),Y4=V2
(10)
We consider the following step functions:
ζj(y)={0for{cj<y<h,j=1,40<y<cj,j=2,31for{0<y<cj,j=1,4cj<y<h,j=2,3
(11)
Let us consider the following transformation:
u(y)=U(y)(X+Y)
(12)
with u(y)={uj(y)}4×1T,X={Xj}4×1T, Y={Yj}4×1T,U(y)={Ujl(y)}4×4.
Utilizing Havelock’s [20] inversion formulae on uj(y) and using continuity of ϕ(x, y) at x = sj, we get a system of integral equations. Equations (10) and (11) are successively used to combine those integral equations in a single one and then convert their ranges of integration into (0, h). After that, the transformation (12) helps to recast the combined integral equation into the following matrix form:
ζ(y)V(y)=0hζ(y)P(y,u)ζ(u)U(u)du,0<y<h
(13)
where ζ(y)=diag(ζj(y))4×4,V(y)=diag(coshk0(hy)/coshk0h; coshα0(cy)/coshα0c,coshα0(cy)/coshα0c,coshk0(hy)/coshk0h) and
P(y,u)=(P1(y,u)P2(y,u)00P3(y,u)P4(y,u)P5(y,u)00P6(y,u)P7(y,u)P8(y,u)00P9(y,u)P10(y,u))
Here, the components of P(y, u), i.e., Pj(y,u)(j=1,2,10) are given by
P1(y,u)=4α0cotq0(s2s1)coshα0(cu)coshα0(cy)q0(2α0c+sinh2α0c)n=14αncothqn(s2s1)cosαn(cu)cosαn(cy)qn(2αnc+sin2αnc)n=14kncoskn(hu)coskn(hy)pn(2knh+sin2knh)
(14a)
P2(y,u)=4α0cscq0(s2s1)coshα0(cu)coshα0(cy)q0(2α0c+sinh2α0c)+n=14αncschqn(s2s1)cosαn(cu)cosαn(cy)qn(2αnc+sin2αnc)
(14b)
P3(y,u)=P2(y,u)
(14c)
P4(y,u)=4α0{2csc2q0(s3s2)+cotq0(s2s1)}coshα0(cu)coshα0(cy)q0(2α0c+sinh2α0c)n=14αn{cothqn(s2s1)+cothqn(s3s2)}cosαn(cu)cosαn(cy)qn(2αnc+sin2αnc)
(14d)
P5(y,u)=4α0cscq0(s3s2)coshα0(cu)coshα0(cy)q0(2α0c+sinh2α0c)+n=14αncschqn(s3s2)cosαn(cu)cosαn(cy)qn(2αnc+sin2αnc)
(14e)
P6(y,u)=P5(y,u)
(14f)
P7(y,u)=4α0{2csc2q0(s4s3)+cotq0(s3s2)}coshα0(cu)coshα0(cy)q0(2α0c+sinh2α0c)n=14αn{cothqn(s4s3)+cothqn(s3s2)}cosαn(cu)cosαn(cy)qn(2αnc+sin2αnc)
(14g)
P8(y,u)=4α0cscq0(s4s3)coshα0(cu)coshα0(cy)q0(2α0c+sinh2α0c)+n=14αncschqn(s4s3)cosαn(cu)cosαn(cy)qn(2αnc+sin2αnc)
(14h)
P9(y,u)=P8(y,u)
(14i)
P10(y,u)=4α0cotq0(s4s3)coshα0(cu)coshα0(cy)q0(2α0c+sinh2α0c)n=14αncothqn(s4s3)cosαn(cu)cosαn(cy)qn(2αnc+sin2αnc)n=14kncoskn(hu)coskn(hy)pn(2knh+sin2knh)
(14j)
Again, using Havelock’s [20] inversion formulae, the constants V1,T0(3),T0(4), and V2 are expressed in terms of horizontal components of velocities. Thereafter, utilizing Eqs. (10)(12), we obtain the following matrix equation:
Y=ZX
(15)
where Z=[WS+Q]1[WSQ],W=diag((2k0h+sinh2k0h)/4cosh2k0h,(2α0c+sinh2α0c)/4cosh2α0c,(2α0c+sinh2α0c)4cosh2α0c), (2k0h+sinh2k0h)/4cosh2k0h)S=diag(iκ/k0q0, cotq0(s3s2)/α0,q0cotq0(s4s3)/α0,iκ/k0) and Q is of the form
Q=0hζ(u)V(u)U(u)du
(16)
In order to find Z, the elements V1 and V2 of Y are determined by solving Eq. (15). Thus, reflection and transmission coefficients R and T, connected to V1 and V2, respectively, are obtained.
Now, we consider (N + 1)-term Galerkin’s approximation of U(u) to solve the integral equations as given by
Ujl(u)n=0Nbjl(n)χj(n)(u),j,l=1,2,3,4
(17)
where bjl(n) are unknown constants and χj(n)(u) the basis functions.
On substituting Eq. (17) into Eq. (13), we obtain four integral equations. Multiplying jth integral equation by χj(m)(u)(j=1,2,3,4) and integrating over (0, h) successively, the following system of equation is obtained as
n=0Nj=14bjl(n)Kij(mn)=δilLi(m),i,l=1,2,3,4;m=0,1,,N
(18)
where
K11(mn)=0c1χ1(m)(y){0c1P1(y,u)χ1(n)(u)du}dy
(19a)
K12(mn)=0c1χ1(m)(y){c2hP2(y,u)χ2(n)(u)du}dy
(19b)
K13(mn)=0
(19c)
K14(mn)=0
(19d)
K21(mn)=c2hχ2(m)(y){0c1P3(y,u)χ1(n)(u)du}dy
(19e)
K22(mn)=c2hχ2(m)(y){c2hP4(y,u)χ2(n)(u)du}dy
(19f)
K23(mn)=c2hχ2(m)(y){c3hP5(y,u)χ3(n)(u)du}dy
(19g)
K24(mn)=0
(19h)
K31(mn)=0
(19i)
K32(mn)=c3hχ3(m)(y){c2hP6(y,u)χ2(n)(u)du}dy
(19j)
K33(mn)=c3hχ3(m)(y){c3hP7(y,u)χ3(n)(u)du}dy
(19k)
K34(mn)=c3hχ3(m)(y){0c4P8(y,u)χ4(n)(u)du}dy
(19l)
K41(mn)=0
(19m)
K42(mn)=0
(19n)
K43(mn)=0c4χ4(m)(y){c3hP9(y,u)χ3(n)(u)du}dy
(19o)
K44(mn)=0c4χ4(m)(y){0c4P10(y,u)χ4(n)(u)du}dy
(19p)
L1(m)=0c1χ1(m)(y)coshk0(hy)coshk0hdy
(19q)
L2(m)=c2hχ2(m)(y)coshα0(cy)coshα0cdy
(19r)
L3(m)=c3hχ3(m)(y)coshα0(cy)coshα0cdy
(19s)
L4(m)=0c4χ4(m)(y)coshk0(hy)coshk0hdy
(19t)
The basis functions χj(n)(y) ’s are chosen as (cf. Mandal and Chakrabarti [23])
χj(n)(y)=ddy[eKyycjeKuχ^j(n)(u)du],{y(0,cj)(j=1,4)y(cj,h)(j=2,3)
(20)
with
χ^j(n)(u)={27/6Γ(16)(2n)!πΓ(2n+13)cj1/3(cj2t2)1/3C2n1/6(ucj),u(0,cj)(j=1,4)2(1)nπ{(hcj)2(ht)2}1/2T2n(huhcj),u(cj,h)(j=2,3)
(21)
where C2n1/6(x) is the ultra-spherical Gegenbauer polynomial of order 16 and T2n(x) the Chebychev polynomial of the first kind of order 2n.
With the assistance of Pj(y,u)(j=1,2,10) from Eq. (14), Kij(mn),Li(m)(i,j=1,2,3,4;m,n=0,1,N) are of the following form:
K11(mn)=16α0cotq0(s2s1)cosh2α0cq0(2α0c+sinh2α0c)I2m+(1/6)(α0c1)I2n+(1/6)(α0c1)(α0c1)1/3r=116(1)m+nαrcothqr(s2s1)qr(2αrc+sin2αrc)J2m+(1/6)(αrc1)J2n+(1/6)(αrc1)(αrc1)1/3r=116(1)m+nkrcos2krcpr(2krh+sin2krh)J2m+(1/6)(krc1)J2n+(1/6)(krc1)(krc1)1/3
(22a)
K12(mn)=8(1)nα0cscq0(s2s1)coshα0cq0(2α0c+sinh2α0c)I2m+(1/6)(α0c1)(α0c1)1/6I2n{α0(cc2)}+r=18(1)mαrcschqr(s2s1)qr(2αrc+sin2αrc)J2m+(1/6)(αrc1)(αrc1)1/6J2n{αr(cc2)}
(22b)
K13(mn)=0
(22c)
K14(mn)=0
(22d)
K21(mn)=8(1)mα0cscq0(s2s1)coshα0cq0(2α0c+sinh2α0c)I2m{α0(cc2)}I2n+(1/6)(α0c1)(α0c1)1/6+r=18(1)nαrcschqr(s2s1)qr(2αrc+sin2αrc)J2m{αr(cc2)}J2n+(1/6)(αrc1)(αrc1)1/6
(22e)
K22(mn)=4α0(1)m+n{2csc2q0(s3s2)+cotq0(s2s1)}q0(2α0c+sinh2α0c)I2m{α0(cc2)}I2n{α0(cc2)}
(22f)
r=14αr{cothqr(s2s1)+cothqr(s3s2)}qr(2αrc+sin2αrc)J2m{αr(cc2)}J2n{αr(cc2)}
(22g)
K23(mn)=4(1)m+nα0cscq0(s3s2)q0(2α0c+sinh2α0c)I2m{α0(cc2)}I2n{α0(cc3)}+r=14αrcschqr(s3s2)qr(2αrc+sin2αrc)J2m{αr(cc2)}J2n{αr(cc3)}
(22h)
K24(mn)=0
(22i)
K31(mn)=0
(22j)
K32(mn)=4(1)m+nα0cscq0(s3s2)q0(2α0c+sinh2α0c)I2m{α0(cc3)}I2n{α0(cc2)}+r=14αrcschqr(s3s2)qr(2αrc+sin2αrc)J2m{αr(cc3)}J2n{αr(cc2)}
(22k)
K33(mn)=4(1)m+nα0{2csc2q0(s4s3)+cotq0(s3s2)}q0(2α0c+sinh2α0c)I2m{α0(cc3)}I2n{α0(cc3)}r=14αr{cothqr(s4s3)+cothqr(s3s2)}qr(2αrc+sin2αrc)J2m{αr(cc3)}J2n{αr(cc3)}
(22l)
K34(mn)=8(1)mα0cscq0(s4s3)coshα0cq0(2α0c+sinh2α0c)I2m{α0(cc3)}I2n+(1/6)(α0c4)(α0c4)1/6+r=18(1)nαrcschqr(s4s3)qr(2αrc+sin2αrc)J2m{αr(cc3)}J2n+(1/6)(αrc4)(αrc4)1/6
(22m)
K41(mn)=0
(22n)
K42(mn)=0
(22o)
K43(mn)=8(1)nα0cscq0(s4s3)coshα0cq0(2α0c+sinh2α0c)I2m+(1/6)(α0c4)(α0c4)1/6I2n{α0(cc3)}+r=18(1)mαrcschqr(s4s3)qr(2αrc+sin2αrc)J2m+(1/6)(αrc4)(αrc4)1/6J2n{αr(cc3)}
(22p)
K44(mn)=16α0cotq0(s4s3)cosh2α0cq0(2α0c+sinh2α0c)I2m+(1/6)(α0c4)I2n+(1/6)(α0c4)(α0c4)1/3r=116(1)m+nαrcothqr(s4s3)qr(2αrc+sin2αrc)J2m+(1/6)(αrc4)J2n+(1/6)(αrc4)(αrc4)1/3r=116(1)m+nkrcos2krcpr(2krh+sin2krh)J2m+(1/6)(krc4)J2n+(1/6)(krc4)(krc4)1/3
(22q)
L1(m)=2I2m+(1/6)(k0c1)(k0c1)1/6
(22r)
L2(m)=(1)mI2m{α0(cc2)}coshα0c
(22s)
L3(m)=(1)mI2m{α0(cc3)}coshα0c
(22t)
L4(m)=2I2m+(1/6)(k0c4)(k0c4)1/6
(22u)
where Jm,Im are the Bessel function and modified Bessel function of order m, respectively.
Now bjl(n)’s are obtained from Eq. (18) by substituting the values of Kij(mn),Lj(m) (i, j = 1, 2, 3, 4). After that assuming Q = {Qjl}4×4, we get Qjl’s (j, l = 1, 2, 3, 4) from Eq. (16) given below
Qjln=0Nbjl(n)Lj(n),j,l=1,2,3,4
(23)

Validation of the Numerical Model

In this section, the present numerical model is validated with the model of Das et al. [21] for two partially immersed thin vertical barrier as a special limiting case. For this purpose, the dimensionless trench height i.e., c1/h (= c4/h) is taken to be 0.999 so that the bathymetry of the ocean becomes almost like flat rigid bed. In Table 1, the values of |R| of present study are compared with the values of |R1| and |R2| of Das et al. [21] for c2/h = 0.2 ( = c3/h), s1/h = 0.001, s2/h = 0.3, s3/h = 0.9, s4/h = 2.0. It is observed from Table 1 that an agreement up to three decimal places is achieved for three different angles of incidence α = 0 deg, 30 deg, and 60 deg. This assures that the present breakwater model is a valid one and can be considered into various ocean engineering problems.

Table 1

Comparison among the numerical results of reflection coefficient obtained by Das et al. [21] and present results for c1/h = 0.999 (= c4/h), c2/h = 0.2 (= c3/h), s1/h = 0.001, s2/h = 0.3, s3/h = 0.9, s4/h = 2.0

α = 0 degα = 30 degα = 60 deg
Das et al. [21]Present resultDas et al. [21]Present resultDas et al. [21]Present result
KhR1(lowerboundfor|R|)R2(upperboundfor|R|)|R|R1(lowerboundfor|R|)R2(upperboundfor|R|)|R|R1(lowerboundfor|R|)R2(upperboundfor|R|)|R|
0.20.0317990.0323410.0320560.0278430.0283160.0282420.0164270.0167050.016681
0.80.0884450.0890490.0886230.0818990.0824760.0824380.0536740.0538160.053799
1.40.1261880.1288960.1275660.1371670.1391590.1382310.1131300.1139310.113222
α = 0 degα = 30 degα = 60 deg
Das et al. [21]Present resultDas et al. [21]Present resultDas et al. [21]Present result
KhR1(lowerboundfor|R|)R2(upperboundfor|R|)|R|R1(lowerboundfor|R|)R2(upperboundfor|R|)|R|R1(lowerboundfor|R|)R2(upperboundfor|R|)|R|
0.20.0317990.0323410.0320560.0278430.0283160.0282420.0164270.0167050.016681
0.80.0884450.0890490.0886230.0818990.0824760.0824380.0536740.0538160.053799
1.40.1261880.1288960.1275660.1371670.1391590.1382310.1131300.1139310.113222

In order to validate the present results with the results of Mandal and Kanoria [15], we take c2/h = 0.001 = c3/h so that the pair of thin barriers almost vanishes. Thus, the graph of Fig. 6 in Mandal and Kanoria [15] for a thick bottom-standing barrier has been almost recovered in Fig. 2 by taking c1/h = 0.5 ( = c4/h), s1/h = 0.001, s2/h = 0.0011, s3/h = 0.00111, s4/h = 0.2, α = 45 deg. This assures that the numerical method adopted in the present article yields fairly accurate results.

Fig. 2
Graph of |R| versus Kh for c2/h = 0.001 (= c3/h), c1/h = 0.5 (= c4/h), s1/h = 0.001, s2/h = 0.0011, s3/h = 0.00111, s4/h = 0.2, α = 45 deg
Fig. 2
Graph of |R| versus Kh for c2/h = 0.001 (= c3/h), c1/h = 0.5 (= c4/h), s1/h = 0.001, s2/h = 0.0011, s3/h = 0.00111, s4/h = 0.2, α = 45 deg
Close modal

In Table 2, |R| and |T| are tabulated for a pair of partially immersed thin vertical barriers just above a symmetrical trench with parameters c1/h = 0.7 (= c4/h), c2/h = 0.2 (= c3/h), s1/h = 0.1, s2/h = 0.3, s3/h = 0.6, s4/h = 0.9, α = 45 deg. The last column of Table 2 affirms that the energy balance relation (i.e., |R|2+|T|2=1) is verified. This gives the correctness of the present breakwater configuration.

Table 2

Validation of energy balance relation for present numerical results

Kh|R||T||R|2 + |T|2
0.250.1700010.9854441
0.500.2238190.9746311
0.750.2584020.9660371
1.000.2858090.9582871
1.250.3123300.9499741
1.500.3420590.9396781
1.750.3766370.9263611
2.000.4133780.9105591
Kh|R||T||R|2 + |T|2
0.250.1700010.9854441
0.500.2238190.9746311
0.750.2584020.9660371
1.000.2858090.9582871
1.250.3123300.9499741
1.500.3420590.9396781
1.750.3766370.9263611
2.000.4133780.9105591

Some infinite series sum appears in the numerical computations and needs to truncate the series at some finite stage without hampering the convergence of the numerical results. So, our aim is to fix the truncation size and ensure the convergence of the infinite series sums. For that reason, Table 3 is tabulated with different parametric values as c1/h = 0.65 ( = c4/h), c2/h = 0.35 ( = c3/h), s1/h = 1.1, s2/h = 2.2, s3/h = 3.3, s4/h = 4.4, α = 60 deg. Table 3 demonstrates that the reflection coefficients converge exactly five decimal places by fixing truncation size r = 100 in the infinite series sums. Thus, for the sake of convergence of the values of |R| and |T|, truncation size of infinite series sums is chosen judiciously with r = 100 in the subsequent numerical computations. This gives a partial check on the exactness of the present method.

Table 3

Convergence of infinite series sums for different truncation sizes r

Khr = 20r = 25r = 50r = 75r = 100
0.20.3471040.3482250.3485730.3485820.348584
0.40.3985550.3997890.3999010.3999450.399947
0.60.3971690.3983670.3986660.3986770.398679
0.80.3670290.3679450.3680910.3681150.368118
1.00.3135260.3140850.3142550.3142640.314265
Khr = 20r = 25r = 50r = 75r = 100
0.20.3471040.3482250.3485730.3485820.348584
0.40.3985550.3997890.3999010.3999450.399947
0.60.3971690.3983670.3986660.3986770.398679
0.80.3670290.3679450.3680910.3681150.368118
1.00.3135260.3140850.3142550.3142640.314265

Numerical Results and Analysis

In this section, a computer program in Mathematica software has been developed based on the theory discussed in the aforementioned sections and numerical simulations are performed with the different nondimensional parametric values. To understand the effect of various physical parameters like angle of incidence, height of thick barrier, length of thin barrier, width of thick barrier, etc., on wave transformation due to thin and thick barriers, few results of reflection and transmission coefficients are graphically explored. These numerical estimates of |R| and |T| will provide global information on the amount of wave energy reflected back by thick and thin barriers and transmitted on the lee side of the thick barrier which gives a quick assessment on wave transformation by the thick and thin barriers.

Figure 3 presents the effect of the angle of incidence on the wave reflection and transmission. Four graphs are plotted for |R| and |T| against wave number with four different angles of incidence α = 0 deg, 30 deg, 45 deg, and 60 deg. The other dimensionless parameters are taken as c1/h = 0.65 (= c4/h), c2/h = 0.15 (= c3/h), s1/h = 0.25, s2/h = 0.5, s3/h = 0.75, s4/h = 1.0. Figure 3 shows that more waves are reflected and less waves are transmitted with a decrease in the angle of incidence.

Fig. 3
(a) |R| and (b) |T| versus Kh for c1/h = 0.65 (= c4/h), c2/h = 0.15 (= c3/h), s1/h = 0.25, s2/h = 0.5, s3/h = 0.75, s4/h = 1.0
Fig. 3
(a) |R| and (b) |T| versus Kh for c1/h = 0.65 (= c4/h), c2/h = 0.15 (= c3/h), s1/h = 0.25, s2/h = 0.5, s3/h = 0.75, s4/h = 1.0
Close modal

The influence of the height of the thick barrier is revealed in Fig. 4. Under the consideration of the parametric values c2/h = 0.3 (= c3/h), s1/h = 1.0, s2/h = 2.0, s3/h = 3.0, s4/h = 4.0, α = 45 deg, reflection and transmission coefficients are depicted against Kh in Figs. 4(a) and 4(b). It is observed from these figures that the global maxima of |R| increase and the global minima of |T| decrease as the height of the thick bottom-standing barrier increases. It is also noted from Fig. 4(a) that peak values of |R| have the tendency to shift to the right as the height of the thick barrier increases.

Fig. 4
(a) |R| and (b)|T| versus Kh for c2/h = 0.3 (= c3/h), s1/h = 1.0, s2/h = 2.0, s3/h = 3.0, s4/h = 4.0, α = 45 deg
Fig. 4
(a) |R| and (b)|T| versus Kh for c2/h = 0.3 (= c3/h), s1/h = 1.0, s2/h = 2.0, s3/h = 3.0, s4/h = 4.0, α = 45 deg
Close modal

In Fig. 5, |R| and |T| are plotted against Kh with various nondimensional parameters c1/h = 0.6 ( = c4/h), s1/h = 0.6, s2/h = 1.2, s3/h = 1.4, s4/h = 1.8, α = 60° to analyze the impact of length of the pair of thin barriers. Figure 5(a) exhibits that the amount of reflection increases with the length of barriers. On the other hand, Fig. 5(b) manifests totally contrasting nature for the transmission coefficient. This phenomenon occurs due to larger barrier length helps more waves to reflect and less to transmit.

Fig. 5
(a) |R| and (b) |T| versus Kh for c1/h = 0.6 (= c4/h), s1/h = 0.6, s2/h = 1.2, s3/h = 1.4, s4/h = 1.8, α = 60 deg
Fig. 5
(a) |R| and (b) |T| versus Kh for c1/h = 0.6 (= c4/h), s1/h = 0.6, s2/h = 1.2, s3/h = 1.4, s4/h = 1.8, α = 60 deg
Close modal

Curves for absolute estimation of |R| and |T| are shown in Fig. 6 for different separation lengths of the thin barriers. A set of the dimensionless parameters is chosen as c1/h = 0.65 ( = c4/h), c2/h = 0.35 ( = c3/h), s1/h = 0.25, s4/h = 3.25, α = 30°. From Fig. 6(a), it is noteworthy to mention that the spikes of |R| increase as the barrier spacing of thin barriers increases. Figure 6(b) infers that minima of the spikes of |T| decrease when the separation lengths of the thin barrier increase.

Fig. 6
(a) |R| and (b) |T| versus Kh for c1/h = 0.65 (= c4/h), c2/h = 0.35 (= c3/h), s1/h = 0.25, s4/h = 3.25, α = 30 deg
Fig. 6
(a) |R| and (b) |T| versus Kh for c1/h = 0.65 (= c4/h), c2/h = 0.35 (= c3/h), s1/h = 0.25, s4/h = 3.25, α = 30 deg
Close modal

The width of the thick bottom-standing barrier has made a severe impact on reflection and transmission coefficients as shown in Fig. 7. |R| and |T| are both graphically visualized against wave number Kh for three different barrier widths. For this graphical depiction, the nondimensional parameters are considered as c1/h = 0.8 ( = c4/h), c2/h = 0.4 ( = c3/h), s2/h = 2.0, s3/h = 2.5, α = 60 deg. From both Figs. 7(a) and 7(b), it is concluded that some spikes are observed at some discrete frequencies for both |R| and |T|. The maxima of the spikes of |R| and minima of the spikes of |T| have the tendency to shift to the left as the width of the thick barrier increases. Moreover, an interesting fact has been discovered in Figs. 4, 6, and 7. The magnitudes of reflection and transmission coefficients have sudden spikes at some certain frequencies that may be termed as “resonant frequencies.” This is mainly due to the resonant effects in the presence of multiple obstacles in the form of two thin partially immersed vertical barriers and a bottom-standing rectangular thick barrier in a same-barrier configuration.

Fig. 7
(a) |R| and (b) |T| versus Kh for c1/h = 0.8 (= c4/h), c2/h = 0.4 (= c3/h), s2/h = 2.0, s3/h = 2.5, α = 60 deg
Fig. 7
(a) |R| and (b) |T| versus Kh for c1/h = 0.8 (= c4/h), c2/h = 0.4 (= c3/h), s2/h = 2.0, s3/h = 2.5, α = 60 deg
Close modal

Conclusion

This paper presents a mathematical treatment to put a light on wave diffraction by pair of thin vertical partially immersed barriers just above a thick bottom-standing barrier. The prime objective of this study is to address both one-third and half singularities that arise in the integral equations and to provide a promising procedure for handling such complicated numerical calculations. The analytical expressions giving reflection and transmission coefficients are shown graphically against wave number to analyze the effects of various physical parameters. The present results are reproduced and validated with the known results of Das et al. [21] and Mandal and Kanoria [15] in the literature to verify the accuracy of this methodology. From the analysis of illustrations, it is pointed out that the angle of incidence has a crucial impact on the magnitude of the reflection and transmission coefficients. It is observed from the graphical depictions that the global maxima of |R| and global minima of |T| are varying with the height thick bottom-standing barrier. When the height of bottom-standing barrier increases, then peaks of |R| have a propensity to shift to the right. Increasing the length of thin barriers produces more reflection consequently, less transmission to the seaside, yielding less impact on the seashore by a pair of thin vertical partially immersed barriers if imposed at the free surface. Both separation lengths of thin barriers and width of thick barrier have made a severe impact on maxima of peaks of |R| and minima of peaks of |T|. From all these curves, one can predict the wave numbers in which maximum reflection and minimum transmission occur. Some resonant frequencies are also observed in some figures due to multiple obstacles in the form of thin and thick barriers.

This study will be useful to solve integral equations having one-third as well as half singularities which arise in the other areas of mathematical physics. This investigation will be of immense help in the prediction of offshore inundation during extreme events like storm surges and tsunamis. Finally, the authors believe that the present work may provide some interesting information for experimental scientists/marine researchers/seismologists working on this relevant topic.

Acknowledgment

The authors are very much indebted to the reviewers for their valuable suggestions which enabled the authors to carry out the desired revision of the manuscript. This work is partially supported by a SERB, DST(TAR/2022/000107).

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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